reserve x,y for set;
reserve s,r for Real;
reserve r1,r2 for Real;
reserve n for Nat;
reserve p,q,q1,q2 for Point of TOP-REAL 2;
reserve C for Simple_closed_curve;

theorem Th67:
  Upper_Middle_Point C in Upper_Arc C
proof
  set L = Vertical_Line((W-bound C+E-bound C)/2),
  p = First_Point(Upper_Arc C,W-min C,E-max C,L);
A1: Upper_Arc C meets L by Th63;
  L is closed by Th6;
  then
A2: Upper_Arc C /\ L is closed by TOPS_1:8;
A3: Upper_Arc C is_an_arc_of E-max C, W-min C by Th50;
  then p = Last_Point(Upper_Arc C,E-max C,W-min C,L) by A1,A2,JORDAN5C:18;
  then p in L /\ Upper_Arc C by A1,A2,A3,JORDAN5C:def 2;
  hence thesis by XBOOLE_0:def 4;
end;
